Theorem dfac5 10112

Description: Equivalence of two versions of the Axiom of Choice. The right-hand side is Theorem 6M(4) of [Enderton] p. 151 and asserts that given a family of mutually disjoint nonempty sets, a set exists containing exactly one member from each set in the family. The proof does not depend on AC. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.)

Assertion

Ref	Expression
dfac5	⊢ (CHOICE ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))

Distinct variable group:   𝑥,𝑧,𝑦,𝑤,𝑣

Proof of Theorem dfac5

Dummy variables 𝑓 ℎ 𝑢 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfac4 10106	. . 3 ⊢ (CHOICE ↔ ∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)))
2		neeq1 3026	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → (𝑧 ≠ ∅ ↔ 𝑤 ≠ ∅))
3	2	cbvralvw 3249	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ↔ ∀𝑤 ∈ 𝑥 𝑤 ≠ ∅)
4	3	anbi2i 634	. . . . . . . . . . 11 ⊢ ((∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑧 ∈ 𝑥 𝑧 ≠ ∅) ↔ (∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑤 ∈ 𝑥 𝑤 ≠ ∅))
5		r19.26 3131	. . . . . . . . . . 11 ⊢ (∀𝑤 ∈ 𝑥 ((𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) ↔ (∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑤 ∈ 𝑥 𝑤 ≠ ∅))
6	4, 5	bitr4i 281	. . . . . . . . . 10 ⊢ ((∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑧 ∈ 𝑥 𝑧 ≠ ∅) ↔ ∀𝑤 ∈ 𝑥 ((𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅))
7		pm3.35 814	. . . . . . . . . . . 12 ⊢ ((𝑤 ≠ ∅ ∧ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)) → (𝑓‘𝑤) ∈ 𝑤)
8	7	ancoms 463	. . . . . . . . . . 11 ⊢ (((𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) → (𝑓‘𝑤) ∈ 𝑤)
9	8	ralimi 3108	. . . . . . . . . 10 ⊢ (∀𝑤 ∈ 𝑥 ((𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) → ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤)
10	6, 9	sylbi 220	. . . . . . . . 9 ⊢ ((∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑧 ∈ 𝑥 𝑧 ≠ ∅) → ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤)
11		r19.26 3131	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) ↔ (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)))
12		elin 3929	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ ran 𝑓))
13		fvelrnb 6942	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 Fn 𝑥 → (𝑣 ∈ ran 𝑓 ↔ ∃𝑡 ∈ 𝑥 (𝑓‘𝑡) = 𝑣))
14	13	biimpac 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑣 ∈ ran 𝑓 ∧ 𝑓 Fn 𝑥) → ∃𝑡 ∈ 𝑥 (𝑓‘𝑡) = 𝑣)
15		fveq2 6882	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = 𝑡 → (𝑓‘𝑤) = (𝑓‘𝑡))
16		id 23	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = 𝑡 → 𝑤 = 𝑡)
17	15, 16	eleq12d 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 = 𝑡 → ((𝑓‘𝑤) ∈ 𝑤 ↔ (𝑓‘𝑡) ∈ 𝑡))
18		neeq2 3027	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = 𝑡 → (𝑧 ≠ 𝑤 ↔ 𝑧 ≠ 𝑡))
19		ineq2 4175	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = 𝑡 → (𝑧 ∩ 𝑤) = (𝑧 ∩ 𝑡))
20	19	eqeq1d 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = 𝑡 → ((𝑧 ∩ 𝑤) = ∅ ↔ (𝑧 ∩ 𝑡) = ∅))
21	18, 20	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 = 𝑡 → ((𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)))
22	17, 21	anbi12d 643	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑤 = 𝑡 → (((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) ↔ ((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅))))
23	22	rspcv 3586	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 ∈ 𝑥 → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅))))
24		minel 4432	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ∩ 𝑡) = ∅) → ¬ (𝑓‘𝑡) ∈ 𝑧)
25	24	ex 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓‘𝑡) ∈ 𝑡 → ((𝑧 ∩ 𝑡) = ∅ → ¬ (𝑓‘𝑡) ∈ 𝑧))
26	25	imim2d 58	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓‘𝑡) ∈ 𝑡 → ((𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅) → (𝑧 ≠ 𝑡 → ¬ (𝑓‘𝑡) ∈ 𝑧)))
27	26	imp 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) → (𝑧 ≠ 𝑡 → ¬ (𝑓‘𝑡) ∈ 𝑧))
28	27	necon4ad 2983	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) → ((𝑓‘𝑡) ∈ 𝑧 → 𝑧 = 𝑡))
29		eleq1 2857	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓‘𝑡) = 𝑣 → ((𝑓‘𝑡) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧))
30	29	biimpar 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓‘𝑡) = 𝑣 ∧ 𝑣 ∈ 𝑧) → (𝑓‘𝑡) ∈ 𝑧)
31	28, 30	impel 514	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) ∧ ((𝑓‘𝑡) = 𝑣 ∧ 𝑣 ∈ 𝑧)) → 𝑧 = 𝑡)
32		fveq2 6882	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = 𝑡 → (𝑓‘𝑧) = (𝑓‘𝑡))
33		eqeq2 2781	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓‘𝑡) = 𝑣 → ((𝑓‘𝑧) = (𝑓‘𝑡) ↔ (𝑓‘𝑧) = 𝑣))
34		eqcom 2776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓‘𝑧) = 𝑣 ↔ 𝑣 = (𝑓‘𝑧))
35	33, 34	bitrdi 290	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓‘𝑡) = 𝑣 → ((𝑓‘𝑧) = (𝑓‘𝑡) ↔ 𝑣 = (𝑓‘𝑧)))
36	32, 35	imbitrid 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓‘𝑡) = 𝑣 → (𝑧 = 𝑡 → 𝑣 = (𝑓‘𝑧)))
37	36	ad2antrl 740	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) ∧ ((𝑓‘𝑡) = 𝑣 ∧ 𝑣 ∈ 𝑧)) → (𝑧 = 𝑡 → 𝑣 = (𝑓‘𝑧)))
38	31, 37	mpd 16	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) ∧ ((𝑓‘𝑡) = 𝑣 ∧ 𝑣 ∈ 𝑧)) → 𝑣 = (𝑓‘𝑧))
39	38	exp32 425	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) → ((𝑓‘𝑡) = 𝑣 → (𝑣 ∈ 𝑧 → 𝑣 = (𝑓‘𝑧))))
40	23, 39	syl6com 38	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑡 ∈ 𝑥 → ((𝑓‘𝑡) = 𝑣 → (𝑣 ∈ 𝑧 → 𝑣 = (𝑓‘𝑧)))))
41	40	com14 97	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 ∈ 𝑧 → (𝑡 ∈ 𝑥 → ((𝑓‘𝑡) = 𝑣 → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → 𝑣 = (𝑓‘𝑧)))))
42	41	rexlimdv 3170	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 ∈ 𝑧 → (∃𝑡 ∈ 𝑥 (𝑓‘𝑡) = 𝑣 → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → 𝑣 = (𝑓‘𝑧))))
43	14, 42	syl5 35	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 ∈ 𝑧 → ((𝑣 ∈ ran 𝑓 ∧ 𝑓 Fn 𝑥) → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → 𝑣 = (𝑓‘𝑧))))
44	43	expd 420	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ 𝑧 → (𝑣 ∈ ran 𝑓 → (𝑓 Fn 𝑥 → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → 𝑣 = (𝑓‘𝑧)))))
45	44	com4t 94	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 Fn 𝑥 → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑣 ∈ 𝑧 → (𝑣 ∈ ran 𝑓 → 𝑣 = (𝑓‘𝑧)))))
46	45	imp4b 426	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))) → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ ran 𝑓) → 𝑣 = (𝑓‘𝑧)))
47	12, 46	biimtrid 245	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓‘𝑧)))
48	11, 47	sylan2br 606	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 Fn 𝑥 ∧ (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓‘𝑧)))
49	48	anassrs 472	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓‘𝑧)))
50	49	adantlr 727	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓‘𝑧)))
51		fveq2 6882	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑧 → (𝑓‘𝑤) = (𝑓‘𝑧))
52		id 23	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑧 → 𝑤 = 𝑧)
53	51, 52	eleq12d 2863	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑧 → ((𝑓‘𝑤) ∈ 𝑤 ↔ (𝑓‘𝑧) ∈ 𝑧))
54	53	rspcv 3586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ 𝑥 → (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 → (𝑓‘𝑧) ∈ 𝑧))
55		fnfvelrn 7076	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓 Fn 𝑥 ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ ran 𝑓)
56	55	expcom 418	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ 𝑥 → (𝑓 Fn 𝑥 → (𝑓‘𝑧) ∈ ran 𝑓))
57	54, 56	anim12d 620	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝑥 → ((∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 ∧ 𝑓 Fn 𝑥) → ((𝑓‘𝑧) ∈ 𝑧 ∧ (𝑓‘𝑧) ∈ ran 𝑓)))
58		elin 3929	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓) ↔ ((𝑓‘𝑧) ∈ 𝑧 ∧ (𝑓‘𝑧) ∈ ran 𝑓))
59	57, 58	imbitrrdi 255	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝑥 → ((∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 ∧ 𝑓 Fn 𝑥) → (𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓)))
60	59	expd 420	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑥 → (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 → (𝑓 Fn 𝑥 → (𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓))))
61	60	com13 89	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 Fn 𝑥 → (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 → (𝑧 ∈ 𝑥 → (𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓))))
62	61	imp31 422	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓))
63		eleq1 2857	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑓‘𝑧) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ (𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓)))
64	62, 63	syl5ibrcom 250	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) → (𝑣 = (𝑓‘𝑧) → 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
65	64	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑣 = (𝑓‘𝑧) → 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
66	50, 65	impbid 215	. . . . . . . . . . . . . 14 ⊢ ((((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧)))
67	66	ex 417	. . . . . . . . . . . . 13 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) → (∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧))))
68	67	alrimdv 1956	. . . . . . . . . . . 12 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) → (∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧))))
69		fvex 6895	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑧) ∈ V
70		eqeq2 2781	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑓‘𝑧) → (𝑣 = ℎ ↔ 𝑣 = (𝑓‘𝑧)))
71	70	bibi2d 345	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑓‘𝑧) → ((𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ℎ) ↔ (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧))))
72	71	albidv 1947	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝑓‘𝑧) → (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ℎ) ↔ ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧))))
73	69, 72	spcev 3574	. . . . . . . . . . . . 13 ⊢ (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧)) → ∃ℎ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ℎ))
74		eu6 2608	. . . . . . . . . . . . 13 ⊢ (∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ ∃ℎ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ℎ))
75	73, 74	sylibr 237	. . . . . . . . . . . 12 ⊢ (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧)) → ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))
76	68, 75	syl6 36	. . . . . . . . . . 11 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) → (∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
77	76	ralimdva 3183	. . . . . . . . . 10 ⊢ ((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) → (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
78	77	ex 417	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑥 → (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 → (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))))
79	10, 78	syl5 35	. . . . . . . 8 ⊢ (𝑓 Fn 𝑥 → ((∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑧 ∈ 𝑥 𝑧 ≠ ∅) → (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))))
80	79	expd 420	. . . . . . 7 ⊢ (𝑓 Fn 𝑥 → (∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) → (∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ → (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))))
81	80	imp4b 426	. . . . . 6 ⊢ ((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)) → ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
82		vex 3467	. . . . . . . 8 ⊢ 𝑓 ∈ V
83	82	rnex 7907	. . . . . . 7 ⊢ ran 𝑓 ∈ V
84		ineq2 4175	. . . . . . . . . 10 ⊢ (𝑦 = ran 𝑓 → (𝑧 ∩ 𝑦) = (𝑧 ∩ ran 𝑓))
85	84	eleq2d 2855	. . . . . . . . 9 ⊢ (𝑦 = ran 𝑓 → (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
86	85	eubidv 2620	. . . . . . . 8 ⊢ (𝑦 = ran 𝑓 → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
87	86	ralbidv 3194	. . . . . . 7 ⊢ (𝑦 = ran 𝑓 → (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
88	83, 87	spcev 3574	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
89	81, 88	syl6 36	. . . . 5 ⊢ ((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)) → ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
90	89	exlimiv 1957	. . . 4 ⊢ (∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)) → ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
91	90	alimi 1838	. . 3 ⊢ (∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)) → ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
92	1, 91	sylbi 220	. 2 ⊢ (CHOICE → ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
93		eqid 2769	. . . . 5 ⊢ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))} = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))}
94		biid 264	. . . . 5 ⊢ (∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
95		eqid 2769	. . . . 5 ⊢ (∪ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))} ∩ 𝑦) = (∪ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))} ∩ 𝑦)
96	93, 94, 95	dfac5lem5 10111	. . . 4 ⊢ (∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) → ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
97	96	alrimiv 1954	. . 3 ⊢ (∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) → ∀ℎ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
98		dfac3 10105	. . 3 ⊢ (CHOICE ↔ ∀ℎ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
99	97, 98	sylibr 237	. 2 ⊢ (∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) → CHOICE)
100	92, 99	impbii 212	1 ⊢ (CHOICE ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃!weu 2602  {cab 2747   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ∩ cin 3912  ∅c0 4294  {csn 4594  ∪ cuni 4876   × cxp 5660  ran crn 5663   Fn wfn 6532  ‘cfv 6537  CHOICEwac 10099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ac 10100

This theorem is referenced by:  dfackm  10150  ac8  10476  dfac5prim  45591